1. Field of the Invention
The present invention relates to a method of gradual deformation of a Boolean model simulating a heterogeneous medium, constrained to dynamic data.
2. Description of the Prior Art
The following documents mentioned in the description hereafter illustrate the state of the art:    Chilès, J. P., and Delfiner, P., 1999, Geostatistics: Modeling Spatial Uncertainty, Wiley, New York, 695p.,    Hu, L.-Y., 2000a, Gradual Deformation and Interative Calibration of Gaussian-Related Stochastic Models, Math. Geol., 32(1),    Hu, L.-Y., 2000b, Gradual Deformation of Non-Gaussian Stochastic Models, Geostats 2000 Cape Town, W J Kleingeld and D G Krige (eds.), 1, 94-103,    Hu, L.-Y., 2003, History Matching of Object-Based Stochastic Reservoir Models, SPE 81503,    Journel, A., and Huijbregts, C. J., 1978, Mining Geostatistics, Academic Press, London, 600p.,    Le Ravalec, M., Noetinger, B., and Hu, L.-Y., 2000, The FFT Moving Average (FFT-MA) Generator: An Efficient Numerical Method for Generating and Conditioning GaussianSimulations, Math. Geol., 32(6), 701-723,    Le Ravalec, M., Hu, L.-Y., and Noetinger, B., 2001, Stochastic Reservoir Modeling Constrained to Dynamic Data: Local Calibration and Inference of the Structural Parameters, SPE Journal, 25-31.
Various methods based on a gradual deformation scheme are described in the assignee's French patents 2,780,798, 2,795,841, and 2,821,946, and in French patent applications 02/13,632 or 03/02,199.
Realizations of Gaussian or Gaussian-related stochastic models are often used to represent the spatial distribution of certain physical properties, such as permeability or porosity, in underground reservoirs. Inverse methods are then commonly used to constrain these realizations to data on which these methods depend in a non-linear manner. This is notably the case in hydrology or in the petroleum industry. These methods are based on minimization of an objective function, also referred to as cost function, which measures the difference between the data measured in the field and the corresponding responses numerically simulated for realizations representing the medium to be characterized. The goal is to identify the realizations associated with the lowest objective function values, that is the most coherent realizations as regards the data.
The gradual deformation method was introduced in this context (see Hu, 2000a; Le Ravalec et al., 2000). This geostatistical parameterization technique allows gradual modifications of the realizations from a limited number of parameters. It is particularly well-suited to minimization problems because, when applied to realizations, it induces a continuous and regular variation of the objective function. In fact, minimization can be performed from the most advanced techniques, that is gradient techniques. The gradual deformation method has proved efficient for constraining oil reservoir models to production data (See Le Ravalec et al., 2001).
The gradual deformation method initially set up for Gaussian models has afterwards been extended to non-Gaussian models (See Hu, 2000b) and more particularly to object or Boolean models. These models are used to describe media comprising for example channels or fractures. The channels or fractures are then considered as objects. An algorithm has been proposed to simulate the gradual migration of objects in space, that is the gradual displacement of objects in space. For a realization of an object model, this type of perturbation translates into a smoothed variation of the objective function, as for Gaussian models. This algorithm has then been generalized to the non-stationary and conditional Boolean model (See Hu, 2003). Besides, still using the gradual deformation of Gaussian laws, solutions allowing progressive modification of the number of objects that populate a model are proposed (See Hu, 2003), this number being representative of a Poisson's law. However, one limit of these developments is that the objects appear or disappear suddenly, which can generate severe discontinuities of the objective function and make the gradient techniques conventionally used for carrying out the minimization process difficult to implement.
Gradual Deformation: Reminders
Multi-Gaussian Random Function
The gradual deformation principles that have been proposed to date apply to multi-Gaussian random functions. Let there be, for example, two independent random functions Y1(x) and Y2(x), multi-Gaussian and stationary of order 2 with x being the position vector. These two functions are assumed to have the same means and variances, i.e. 0 and 1, and the same covariance function. A new random function Y(t) is then constructed by combining Y1 and Y2 according to the expression as follows:Y(t)=Y1 cos(t)+Y2 sin(t).
It can be shown that, whatever t, Y has the same mean, variance and covariance model as Y1 and Y2. Besides, Y(t) is also a multi-Gaussian random function because it is the sum of two multi-Gaussian random functions.
According to this combination principle, a chain of realizations y(t) depending only on deformation parameter t can be constructed from two independent realizations y1 and y2 of Y1 and Y2. The basic minimization processes use gradual deformation to explore the chain of realizations and to determine the deformation parameter providing the realization which is the most compatible with the data measured in the field, that is the pressures, production rates, breakthrough times, etc. Since exploration of a single chain of realizations does generally not allow identification of a realization providing a sufficiently small objective function, the desired process is iterated. The optimum realization determined for the 1st chain is then combined with a new independent realization of Y2, and a new chain of realizations whose exploration can provide a realization which reduces the objective function even further is deduced therefrom, etc.
Poisson Point Process
The key element of Boolean models is a Poisson point process which characterizes the spatial layout of the objects. A base Boolean model is considered for which the objects have the same shape and are randomly and uniformly distributed in space. The positions of these objects are distributed according to the Poisson point process of constant density. In other words, the position of an object in space with n dimensions [0,1]n is defined by vector x whose n components are uniform numbers drawn independently according to the uniform distribution law between 0 and 1.
The objects migration technique (See Hu, 2000b) gradually deforms the position of the objects. The position of an object is determined by uniform numbers. First and foremost, these uniform numbers are converted to Gaussian numbers:Y=G−1(x).
G is the standard normal distribution function. Let x1 be the initial position of a given object and x2 another possible position, independent of x1. A trajectory is defined for the object by combining the Gaussian transforms of these two positions according to the gradual deformation method:x(t)=G└G−1(x1)cos(t)+G−1(x2)sin(t)┘
It can be shown that, for any value of deformation parameter t, x is a uniform point of [0,1]n. When the two positions x1 and x2 are fixed, the trajectory is completely determined. A two-dimensional example is shown in FIG. 1.
The object migration technique is a first move towards the gradual deformation of Boolean simulations. One of its limits is that the number of objects is assumed to be constant during deformation. Solutions have been proposed to progressively modify the number of objects that populate a model (See patent application N-01/03,194 or Hu, 2003 mentioned above). However, one limit of these developments is that the objects appear or disappear suddenly, which can generate severe discontinuities of the objective function. The gradual deformation method according to the invention allows, as described below, to reduce this discontinuity and thus to facilitate the implementation of gradient-based optimization techniques.